SK_1 and Lie Algebras

TL;DR

This paper explores conditions under which the algebraic K-theory group SK_1 vanishes for Iwasawa algebras of p-adic Lie groups, linking it to linear algebra problems in Lie algebra theory.

Contribution

It reduces the problem of SK_1 vanishing to a linear algebra problem and solves it for Chevalley orders in split reductive Lie algebras.

Findings

SK_1(A(G)) vanishes under certain conditions

Reduction of the problem to linear algebra over rings

Solution provided for Chevalley orders in split reductive Lie algebras

Abstract

We investigate the vanishing of the group SK1(A(G)) for the Iwasawa algebra A(G) of a pro-p p-adic Lie group G (with p different from 2). We reduce this vanishing to a linear algebra problem for Lie algebras over arbitrary rings, which we solve for Chevalley orders in split reductive Lie algebras.